On the construction of optimal linear codes of dimension four. (English) Zbl 07769988
Summary: A fundamental problem in coding theory is to find \(n_q (k,d)\), the minimum length \(n\) for which an \([n,k,d]_q\) code exists. We show that some \(q\)-divisible optimal linear codes of dimension \(4\) over \(\mathbb{F}_q\), which are not of Belov type, can be constructed geometrically using hyperbolic quadrics in \(\mathrm{PG}(3,q)\). We also construct some new linear codes over \(\mathbb{F}_q\) with \(q=7,8\), which determine \(n_7 (4,d)\) for \(31\) values of \(d\) and \(n_8 (4,d)\) for \(40\) values of \(d\).
References:
| [1] | S. Ball, Table of bounds on three dimensional linear codes or (n, r)-arcs in PG(2, q), available at https://web.mat.upc.edu/people/simeon.michael.ball/codebounds.html |
| [2] | B. I. Belov, V. N. Logachev, and V. P. Sandimirov, Construction of a class of linear binary codes that attain the Varšamov-Griesmer bound, Problemy Peredači Informacii 10 (1974), no. 3, 36-44. · Zbl 0317.94014 |
| [3] | J. Bierbrauer, Introduction to Coding Theory, Chapman & Hall, Dordrecht, 2005. · Zbl 1060.94001 |
| [4] | N. Bono, M. Fujii, and T. Maruta, On optimal linear codes of dimension 4, J. Alge-bra Comb. Discrete Struct. Appl. 8 (2021), no. 2, 73-90. https://doi.org/10.13069/ jacodesmath.935947 · Zbl 1494.94050 · doi:10.13069/jacodesmath.935947 |
| [5] | I. G. Bouyukliev, Y. Kageyama, and T. Maruta, On the minimum length of linear codes over F 5 , Discrete Math. 338 (2015), no. 6, 938-953. https://doi.org/10.1016/j.disc. 2015.01.010 · Zbl 1406.94088 · doi:10.1016/j.disc.2015.01.010 |
| [6] | A. E. Brouwer and M. van Eupen, The correspondence between projective codes and 2-weight codes, Des. Codes Cryptogr. 11 (1997), no. 3, 261-266. https://doi.org/10. 1023/A:1008294128110 · Zbl 0872.94043 · doi:10.1023/A:1008294128110 |
| [7] | E. J. Cheon, On the upper bound of the minimum length of 5-dimensional linear codes, Australas. J. Combin. 37 (2007), 225-232. · Zbl 1114.94019 |
| [8] | B. Csajbók and T. Héger, Double blocking sets of size 3q − 1 in PG(2, q), European J. Combin. 78 (2019), 73-89. https://doi.org/10.1016/j.ejc.2019.01.004 · Zbl 1419.51009 · doi:10.1016/j.ejc.2019.01.004 |
| [9] | S. M. Dodunekov, Optimal linear codes, Doctor Thesis, Sofia, 1985. |
| [10] | M. Grassl, Tables of linear codes and quantum codes, (electronic table, online), http:// www.codetables.de/. |
| [11] | J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop. 4 (1960), 532-542. https://doi.org/10.1147/rd.45.0532 · Zbl 0234.94009 · doi:10.1147/rd.45.0532 |
| [12] | R. Hill, Optimal linear codes, in Cryptography and coding, II (Cirencester, 1989), 75-104, Inst. Math. Appl. Conf. Ser. New Ser., 33, Oxford Univ. Press, New York, 1992. · Zbl 0742.94012 |
| [13] | R. Hill and E. Kolev, A survey of recent results on optimal linear codes, in Combinatorial designs and their applications (Milton Keynes, 1997), 127-152, Chapman & Hall/CRC Res. Notes Math., 403, Chapman & Hall/CRC, Boca Raton, FL, 1999. · Zbl 0916.94009 |
| [14] | J. W. P. Hirschfeld, Finite projective spaces of three dimensions, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. · Zbl 0574.51001 |
| [15] | W. C. Huffman and V. S. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, Cambridge, 2003. https://doi.org/10.1017/CBO9780511807077 · Zbl 1191.94107 · doi:10.1017/CBO9780511807077 |
| [16] | Y. Inoue and T. Maruta, Construction of new Griesmer codes of dimension 5, Finite Fields Appl. 55 (2019), 231-237. https://doi.org/10.1016/j.ffa.2018.10.007 · Zbl 1404.94153 · doi:10.1016/j.ffa.2018.10.007 |
| [17] | Y. Kageyama and T. Maruta, On the geometric constructions of optimal linear codes, Des. Codes Cryptogr. 81 (2016), no. 3, 469-480. https://doi.org/10.1007/s10623-015-0167-2 · Zbl 1396.94109 · doi:10.1007/s10623-015-0167-2 |
| [18] | K. Kumegawa and T. Maruta, Nonexistence of some Griesmer codes over Fq, Discrete Math. 339 (2016), no. 2, 515-521. https://doi.org/10.1016/j.disc.2015.09.030 · Zbl 1356.94093 · doi:10.1016/j.disc.2015.09.030 |
| [19] | K. Kumegawa and T. Maruta, Non-existence of some 4-dimensional Griesmer codes over finite fields, J. Algebra Comb. Discrete Struct. Appl. 5 (2018), no. 2, 101-116. https://doi.org/10.13069/jacodesmath.427968 · Zbl 1423.94151 · doi:10.13069/jacodesmath.427968 |
| [20] | K. Kumegawa, T. Okazaki, and T. Maruta, On the minimum length of linear codes over the field of 9 elements, Electron. J. Combin. 24 (2017), no. 1, Paper No. 1.50, 27 pp. https://doi.org/10.37236/6394 · Zbl 1361.94054 · doi:10.37236/6394 |
| [21] | I. N. Landjev and T. Maruta, On the minimum length of quaternary linear codes of dimension five, Discrete Math. 202 (1999), no. 1-3, 145-161. https://doi.org/10.1016/ S0012-365X(98)00354-9 · Zbl 0934.94018 · doi:10.1016/S0012-365X(98)00354-9 |
| [22] | T. Maruta, On the minimum length of q-ary linear codes of dimension four, Discrete Math. 208/209 (1999), 427-435. https://doi.org/10.1016/S0012-365X(99)00088-6 · Zbl 0957.94038 · doi:10.1016/S0012-365X(99)00088-6 |
| [23] | T. Maruta, On the nonexistence of q-ary linear codes of dimension five, Des. Codes Cryptogr. 22 (2001), 165-177. · Zbl 0985.94036 |
| [24] | T. Maruta, Construction of optimal linear codes by geometric puncturing, Serdica J. Comput. 7 (2013), no. 1, 73-80. · Zbl 1294.94106 |
| [25] | T. Maruta, Griesmer bound for linear codes over finite fields, available at http:// mars39.lomo.jp/opu/griesmer.htm. · Zbl 0906.94018 |
| [26] | T. Maruta and Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun. 5 (2011), no. 3, 505-520. https://doi.org/10.3934/amc.2011.5.505 · Zbl 1250.94071 · doi:10.3934/amc.2011.5.505 |
| [27] | T. Maruta, M. Shinohara, and M. Takenaka, Constructing linear codes from some orbits of projectivities, Discrete Math. 308 (2008), no. 5-6, 832-841. https://doi.org/10. 1016/j.disc.2007.07.045 · Zbl 1143.94019 · doi:10.1016/j.disc.2007.07.045 |
| [28] | M. Takenaka, K. Okamoto, and T. Maruta, On optimal non-projective ternary linear codes, Discrete Math. 308 (2008), no. 5-6, 842-854. https://doi.org/10.1016/j.disc. 2007.07.044 · Zbl 1145.94017 · doi:10.1016/j.disc.2007.07.044 |
| [29] | H. N. Ward, Divisibility of codes meeting the Griesmer bound, J. Combin. Theory Ser. A 83 (1998), no. 1, 79-93. https://doi.org/10.1006/jcta.1997.2864 · Zbl 0913.94026 · doi:10.1006/jcta.1997.2864 |
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